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When we studied the number line in Section [link] we noted that
Each point on the number line corresponds to a real number, and each real number is located at a unique point on the number line.
The $"+"$ and $"-"$ signs now have two meanings:
$+$ can denote the operation of addition or a positive number.
$-$ can denote the operation of subtraction or a negative number.
$-8$ should be read as "negative eight" rather than "minus eight."
$4+(-2)$ should be read as "four plus negative two" rather than "four plus minus two."
$-6+(-3)$ should be read as "negative six plus negative three" rather than "minus six plusminus three."
$-15-(-6)$ should be read as "negative fifteen minus negative six" rather than "minus fifteenminus minus six."
$-5+7$ should be read as "negative five plus seven" rather than "minus five plus seven."
$0-2$ should be read as "zero minus two."
Write each expression in words.
The opposite of a real number is denoted by placing a negative sign directly in front of the number. Thus, if $a$ is any real number, then $-a$ is its opposite. Notice that the letter $a$ is a variable. Thus, $"a"$ need not be positive, and $"-a"$ need not be negative.
If $a$ is a real number, $-a$ is opposite $a$ on the number line and $a$ is opposite $-a$ on the number line.
$-(-a)$ is opposite $-a$ on the number line. This implies that $-(-a)=a$ .
This property of opposites suggests the double-negative property for real numbers.
If
$a=3$ , then
$-a=-3$ and
$-(-a)=-(-3)=3$ .
If
$a=-4$ , then
$-a=-(-4)=4$ and
$-(-a)=a=-4$ .
Find the opposite of each real number.
Suppose that $a$ is a positive number. What type of number is $-a$ ?
If $a$ is positive, $-a$ is negative.
Suppose that $a$ is a negative number. What type of number is $-a$ ?
If $a$ is negative, $-a$ is positive.
Suppose we do not know the sign of the number $m$ . Can we say that $-m$ is positive, negative, or that we do notknow ?
We must say that we do not know.
A number is denoted as positive if it is directly preceded by ____________________ .
a plus sign or no sign at all
A number is denoted as negative if it is directly preceded by ____________________ .
For the following problems, how should the real numbers be read ? (Write in words.)
For the following problems, write the expressions in words.
$0-(-15)$
Rewrite the following problems in a simpler form.
$-[-(-10)]$
$-[-(-15)]$
$-\{-[-(-11)]\}$
$-\{-[-(-14)]\}$
$-[-(2)]$
$-[-(42)]$
$6-(-14)$
$18-(-12)$
$54-(-18)$
$2-(-1)-(-8)$
$24-(-8)-(-13)$
( [link] ) There is only one real number for which ${(5a)}^{2}=5{a}^{2}$ . What is the number?
0
( [link] ) Simplify $(3xy)(2{x}^{2}{y}^{3})(4{x}^{2}{y}^{4})$ .
( [link] ) Simplify ${({a}^{3}{b}^{2}{c}^{4})}^{4}$ .
( [link] ) Simplify ${\left(\frac{4{a}^{2}b}{3x{y}^{3}}\right)}^{2}$ .
$\frac{16{a}^{4}{b}^{2}}{9{x}^{2}{y}^{6}}$
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